Nonlocal symmetries for bilinear equations and their applications

Nonlo cal symmetries for bilinear equatio n s and their applications Xing-Biao Hu 1 , Sen-Y ue Lou 2 , 3 and Xian-Min Qian 4 1 LSEC, ICMSEC, Academ y of Mathematics and Systems Science, Chinese Academ y of Sciences, Beijing 100190 , China 2 Departmen t of Ph ysics, Ningb o Univ ersit y , Zhejiang , CHINA 3 Departmen t of Ph ysics, Shanghai Jiao T ong Univ ersit y , Sha nghai, 200240, CHINA 4 Departmen t of Ph ysics, Shaox ing College of Arts and Science s, Shaoxing 312000, CHINA Abstract In th is pap er, nonlocal symmetries for the bilinear KP and bilinear BKP equations are re-stu died. Tw o arbitrary parameters are introduced in th ese nonlocal symmetries by considering gauge in v ariance of the bilinear KP and bilinear BKP equations under the transformation f − → f e ax + by + ct . By expanding these nonlo cal symmetries in p ow ers of eac h o f tw o parameters, w e ha ve d erived tw o typ es of bili near NKP hierarchies and tw o typ es of bilinear NBKP hierarc hies. An impress ive observa tion is that bilinea r p ositiv e and negative KP and BKP h ierarc h ies may b e derived from the same nonlocal symmetries for the KP and BKP equations. Besides, as t wo concrete examples, w e h av e deived bilinear B¨ ac klund transformations for t − 2 -flow of the NKP hierarc hy and t − 1 -flow of the NBKP hierarch y . All these results hav e ma de it clear that more nice integ rable prop erties w ould b e found for these obtained NKP hierarc hies and NBKP hierarchies . Since KP and BKP hierarchies ha ve play ed an essential role in soliton t heory , w e believe that th e bilinear NKP and NBKP hiera rc hies will ha ve their righ t place in this field. Keyw o rds: Nonlo cal symmetry; negativ e Kadomtsev- Petviashvili hier arch y , negative BKP hierarch y , bi- linear for m 1 In tro duction Symmetries a nd co nserv ation laws for diff erential equations a re the cen tra l themes of p er p etual in terest in mathematical physics[1, 2]. With the developmen t of integrable systems and soliton theory , a v arie ty o f nonlo cal symmetries have bee n in tensely inv estiga ted in the literature, one of which is potential symmetries. In this pap er we are concerned with ano ther type of no nlo cal symmetries, that is so-called eig enfunction symmetries [3 ]-[10]. T o b e concrete, let us firs t ta ke the KP eq ua tion as an exa mple to see wha t it means by suc h nonlo cal symmetries. It is known that for the KP equation (4 u t − 6 u u x − u xxx ) x − 3 α 2 u y y = 0 , α 2 = ± 1 , (1) we ha ve the following nonlocal symmetry σ [5] given b y σ = ( ψ ψ ∗ ) x (2) where ψ and ψ ∗ satisfy αψ y + ( ∂ 2 x + u − λ ) ψ = 0 , (3) ( − 4 ψ t + 4 ψ xxx + 3 u x ψ + 6 uψ x − 3 α ( ∂ − 1 x u y ) ψ = 0; (4) − αψ ∗ y + ( ∂ 2 x + u − λ ) ψ ∗ = 0 , (5) ( − 4 ψ ∗ t + 4 ψ ∗ xxx + 3 u x ψ ∗ + 6 u ψ ∗ x + 3 α ( ∂ − 1 x u y ) ψ ∗ = 0 . (6) Based on the observ ation that equations (3) and (4) (o r (5) a nd (6)) constitute a Lax pair for the KP (1), it is natural to call no nlo cal symmetry σ in (2) eigenfunction symmetry . Eigenfunction symmetr ies hav e play ed an impor tant role in the following topics [3 ]-[7],[9]-[16]: 1 • P os itive and negative hierarchies • Symmetry constraints • Soliton equations with sources F or example, in [11], one of authors (Lou) has derived fr o m eigenfunction symmetry in (2) a hierarch y of neg ative KP (NKP) eq uations u t − 2 n − 1 = n X k =0 ( P k Q n − k ) x , (7) ( ∂ 2 x + u + α∂ y ) P k = P k − 1 , k = 0 , 1 , 2 , ..., n (8) ( ∂ 2 x + u − α∂ y ) Q k = Q k − 1 , k = 0 , 1 , 2 , ..., n (9) with P − 1 = Q − 1 = 0 b y expanding ψ and ψ ∗ in the following wa y ψ = n X k =0 ( ∂ 2 x + u + α∂ y ) n − k P n λ k , ψ ∗ = n X k =0 ( ∂ 2 x + u − α∂ y ) n − k Q n λ k . In par ticular, if n = 0, we hav e the first member of the NKP hier arch y u t − 1 = ( P 0 Q 0 ) x , (10) ( ∂ 2 x + u + α∂ y ) P 0 = 0 , (11) ( ∂ 2 x + u − α∂ y ) Q 0 = 0 . (12) Through a Miura tra nsformation, (10)-(12) ma y b e transformed into (2+1)- dimensional sinh-Gor don system[17] [ αφ y t + e − 2 φ ( e 2 φ φ xt ) x ] y = − ( s x e 2 φ ) xx , (13) [ e 2 φ ( φ xt − 1 2 C e 2 φ + 1 2 C e − 2 φ )] x + αe 2 φ φ y t + αe 2 φ ( e 2 φ s ) x = 0 , (14) where α 2 = ± 1 and C is an arbitrar y constant. Some results hav e b een done on (13)-(14) [18]-[2 0] . How ever, any further integrable pr o p erties hav e not b een achieved for the whole NKP hierar ch y (7 )- (9). So it would be of in terest to cons ide r some integrable prop er ies for (7)-(9), o ne o f which is Hirota ’s bilinear for m. A usual way to do so is to find out a suitable dep endent v ariable tr ansformatio n for the potentials in (7 )-(9) and then try to transform (7)-(9) in to bilinear for m. F or example, for the first member (10) -(12) of the NKP hierarch y , g iven the dep endent v a riable tr a nsformation u = 2 (ln f ) xx , P 0 = g /f , Q 0 = h/f , we can transform (10)-(12) into bilinear form D x D t f · f = gh, (15) ( D 2 x + αD y ) g · f = 0 , (16) ( D 2 x + αD y ) f · h = 0 , (17) where Hiro ta bilinear o p e rator D m y D k t is defined b y [21] D n x D m y D k t a · b ≡  ∂ ∂ x − ∂ ∂ x ′  n  ∂ ∂ y − ∂ ∂ y ′  m  ∂ ∂ t − ∂ ∂ t ′  k a ( x, y , t ) b ( x ′ , y ′ , t ′ ) | x ′ = x,y ′ = y ,t ′ = t . In par ticular, if we se t α = i , h = g ∗ in (15)-(17), we ha ve D x D t f · f = | g | 2 , (18) ( D 2 x + iD y ) g · f = 0 (19) which is in the Hietarinta’s list o f c o mplex bilinear equations pa ssing Hirota’s 3-so lito n condition[22]. In particular, if t = y , the system (18) and (19) is a bilinear form for Redekopp equations[23]. In the following, we will no t follow this line to find bilinear form f or (7)-(9). Ins tea d we will first cons ider nonlo cal symmetr ies for blinear equations, s ay bilinear K P and bilinear BKP a nd then deriv ing bilinear negative KP and negativ e BKP hierarchies directly b y expanding such type of no nlo cal symmetries. This idea can b e describ ed using the following diagra m. 2 Nonlo cal sy mmetry of K P Negative KP hierarch y Nonlo cal symmetry of bilinear KP Bilinear form fo r NKP hierarchy ✲ ✲ ✻ ✻ According to this scheme, the pr oblem o f finding bilinear for m for the NKP hiera rch y b ecomes the pro blem of constructing nonlo ca l symmetry for the bilinear KP equation. One of the purp o ses in this pap er is to study nonlo cal symmetrie s for bilinea r equations. W e will consider nonlo ca l sy mmetries for bilinear KP equation and bilinear BKP equation. The se c ond purp ose of the paper is to use suc h nonlo cal symmetries to der ive bilinear forms for p ositive and negative KP and BKP hier archies. It is remar ked that in [9, 1 0] nonlo cal symmetries for the bilinear KP and BKP equatio n hav e been used to consider symmetry constra ints for the KP and BKP hier archies. This pap er is or ganized as follows. In section 2 , we will conside r nonlo cal symmetr ie s with t wo different parameters for the bilinear KP equation and then use these symmetries to generate negativ e and positive KP hierarch y in bilinear f orm. Section 3 is dev oted to c o nsidering nonlocal symmetries with tw o differen t parameters for the bilinear BKP eq uation and then use these symmetr ies to gener ate nega tive and p ositive BKP hier arch y in bilinear form. Conclusions and discussions a re given in sectio n 4. 2 Nonlo cal symmetries for the b ilinear KP equati on and its ap- plication It is known that by the dependent v aria ble transformatio n u = 2(ln f ) xx , the KP equa tion (1) ca n b e transformed in to the bilinea r form ( − 4 D x D t + D 4 x + 3 D 2 y ) f · f = 0 . (20) Here w e hav e c hosen α = 1 for the s ake of conv enience in ca lc ulation. Co ncerning (2 0), we hav e t wo sets of bilinear B¨ ac klund tr a nsformations which are g iven as follows ( D y + D 2 x + µD x − λ ) f · g = 0 , (21) (4 D t + 3 D x D y − D 3 x + 3 µD y − 3 λD x ) f · g = 0 ( 22) and ( D y + D 2 x + µD x − λ ) h · f = 0 , (23) (4 D t + 3 D x D y − D 3 x + 3 µD y − 3 λD x ) h · f = 0 , (24) where λ and µ are ar bitrary par ameters. W e hav e the following result Prop ositi on 1. Biline ar KP e quation (20) has a nonlo c al symmetry g iven by σ = f Z x g h f 2 dx ′ (25) wher e g , h satisfy (21)-(24 ) .That me ans σ given by (25 ) satisfies t he fol lowing symmetry e quation ( − 4 D x D t + 3 D 2 y + D 4 x ) σ · f = 0 . (26) Pr o of. By direct ca lculation. 3 Remark: This kind o f nonlo cal symmetry has appe ared in [9] when µ = 0. In the following, we would lik e to present t wo sets of negative KP hierar chies: Case 1: µ = 0 , g = ∞ X i =0 g i λ i , h = ∞ X i =0 h i λ i , W e hav e one negative KP hierar ch y f t − 2 n − 1 = 1 2 1 n ! f Z x  ∂ n ( gh ) ∂ λ n  | λ =0 f 2 dx ′ (27) ( D y + D 2 x − λ ) f · g = 0 (28) ( D y + D 2 x − λ ) h · f = 0 (29) i.e. D x D t − 2 n − 1 f · f = n X i =0 g i h n − i (30) ( D y + D 2 x ) f · g i = f g i − 1 (31) ( D y + D 2 x ) h i · f = h i − 1 f (32) with g − 1 = h − 1 = 0. Obviously , equations (30)-(32) co nstitute bilinear form for the NKP hierarchy (7 )- (9 ) with α = 1. In considera tio n of the fact that there is only one τ function app ea red in the bilinear fo rm for po sitive KP hier arch y , it is natural to inquire whether w e ma y als o trans fo rm (30)-(32) in to a set of bilinear equations with only one τ function. The answer is affirmative. In the following, through concrete examples, we will show that b y in tro ducing a se q uence of additiona l v a riables m, z 1 , z 2 , · · · , equations (30)-(32) may be transfo rmed into a set of bilinear equations with one τ -function. Example 1: n=0. In this c a se, we set f = f ( m ) , g 0 = f ( m − 1) , h 0 = f ( m + 1), Then t − 1 -flow of the NKP hiera rch y (30)-(32) beco mes D x D t − 1 f · f = e D m f · f , (33) ( D y + D 2 x ) e 1 2 D m f · f = 0 , (34) where Hiro ta ’s bilinear difference op era tor exp( δ D m ) is defined b y exp( δ D m ) a ( m ) · b ( m ) ≡ exp  δ ( ∂ ∂ m − ∂ ∂ m ′ )  a ( m ) b ( m ′ ) | m ′ = m = a ( m + δ ) b ( m − δ ) . Example 2: n=1. In this cas e , we set f = f ( m ) , g 0 = f ( m − 1) , h 0 = f ( m + 1) , g 1 = − f z 1 ( m − 1) , h 1 = f z 1 ( m + 1) . Then t − 3 -flow of the NKP hierarch y (30)-(32) b ecomes D x D t − 3 f · f = D z 1 e D m f · f (35) ( D y + D 2 x ) e 1 2 D m f · f = 0 , (36) D z 1 ( D y + D 2 x ) e 1 2 D m f · f = 2 e 1 2 D m f · f . (37) Example 3: n=2. In this cas e , we set f = f ( m ) , g 0 = f ( m − 1) , h 0 = f ( m + 1) , g 1 = − f z 1 ( m − 1) , h 1 = f z 1 ( m + 1) , g 2 = 1 2 f z 1 z 1 ( m − 1) − f z 2 ( m − 1) , h 2 = 1 2 f z 1 z 1 ( m + 1) + f z 2 ( m + 1) . 4 Then t − 5 -flow of the NKP hierarch y (30)-(32) b ecomes D x D t − 5 f · f = ( 1 2 D 2 z 1 + D z 2 ) e D m f · f (38) ( D y + D 2 x ) e 1 2 D m f · f = 0 , (39) D z 1 ( D y + D 2 x ) e 1 2 D m f · f = 2 e 1 2 D m f · f , (40) ( 1 2 D 2 z 1 + D z 2 )( D y + D 2 x ) e 1 2 D m f · f = D z 1 e 1 2 D m f · f . (41) In gener a l, along this line, we may construct bilinear eq uations with one τ - function for the NKP hierarchy (30)-(32) s tep by step. Case 2: λ = 0 , g = ∞ X i =0 g i µ i , h = ∞ X i =0 h i µ i , In this case, we hav e another NKP hierarch y f t − n − 1 = 1 2 1 n ! f Z x  ∂ n ( gh ) ∂ µ n  | µ =0 f 2 dx ′ (42) ( D y + D 2 x + µD x ) f · g = 0 (43) ( D y + D 2 x + µD x ) h · f = 0 (44) i.e. D x D t − n − 1 f · f = n X i =0 g i h n − i (45) ( D y + D 2 x ) f · g i = − D x f · g i − 1 (46) ( D y + D 2 x ) h i · f = − D x h i − 1 · f (47) with g − 1 = h − 1 = 0 Again, w e ma y cons truct bilinear e quations with one τ -function for the NKP hierarchy (45)-(47) by int ro ducing a sequence o f additional v ariables m, z 1 , z 2 , · · · . Here w e just co nsider tw o simplest exa mples. Example 4: n=0 In this cas e, w e s et f = f ( m ) , g 0 = f ( m − 1) , h 0 = f ( m + 1), Then t − 1 -flow of the NKP hiera rch y (45)-(47) beco mes D x D t − 1 f · f = e D m f · f (48) ( D y + D 2 x ) e 1 2 D m f · f = 0 . (49) which is the s a me as Example 1. Example 5: n =1 In this ca se, we set f = f ( m ) , g 0 = f ( m − 1) , h 0 = f ( m + 1) , g 1 = f z 1 ( m − 1) , h 1 = f z 1 ( m + 1). Then t − 2 -flow of the NKP hiera rch y (45)-(47) beco me D x D t − 2 f · f = D z 1 e D m f · f , (50) ( D y + D 2 x ) e 1 2 D m f · f = 0 , (51) [ D z 1 ( D y + D 2 x ) e 1 2 D m + 2 D x e 1 2 D m ] f · f = 0 . (52) F urthermo re, concerning equations (50)-(52), we ha v e the fo llowing result: Prop ositi on 2. A B¨ acklund tr ansformation fo r (50)-(52) is ( D x e − D m 2 − λe − D m 2 − µe D m 2 ) f · g = 0 , (53) ( D 2 x + D y − 2 λD x ) f · g = 0 , (54) ( D x D z e − D m 2 + µD z e D m 2 − λD z e − D m 2 + γ e D m 2 + e − D m 2 ) f · g = 0 . (55) ( D t − 1 2 µ D z e − D m − γ 4 µ 2 e − D m + ζ ) f · g = 0 (56) wher e λ, µ, γ and ζ ar e arbitr ary c onstants, and z ≡ z 1 , t − 2 ≡ t for sh ort. 5 Pr o of. Let f ( n ) b e a s olution of Eqs. ( 50)-(52). What w e need to pro ve is that the function g satisfying (53)-(56) is another solution of Eqs. (50) and (52), i.e., P 1 ≡ [ D x D t − 2 − D z 1 e D m ] g · g = 0 , (57) P 2 ≡ ( D y + D 2 x ) e 1 2 D m g · g = 0 , (58) P 3 ≡ [ D z 1 ( D y + D 2 x ) e 1 2 D m + 2 D x e 1 2 D m ] g · g = 0 . (59) In analo gy with the proof already g iven in [18], w e know that P 2 = 0 and P 3 = 0 hold. Th us it suffices to show that P 1 = 0. In this regard, by using (A1)-(A6), we hav e − P 1 f 2 = 2 D x ( D t f · g ) · f g − 2 D z cosh( 1 2 D m )( e 1 2 D m f · g ) · ( e − 1 2 D m f · g ) = 2 D x ( D t f · g ) · f g − 2 µ D z cosh( 1 2 D m )( D x e − 1 2 D m f · g ) · ( e − 1 2 D m f · g ) = 2 D x ( D t f · g ) · f g + 2 µ D x f g · ( D z e − D m f · g ) − 1 µ D x [( D z f · g ) · ( e − D m f · g ) + f g · ( D z e − D m f · g )] = 2 D x [( D t − 1 µ D z e − D m ) f · g ] · f g − 2 µ sinh( 1 2 D m )[( D x D z e − 1 2 D m f · g ) · ( e − 1 2 D m f · g ) + ( D x e − 1 2 D m f · g ) · ( D z e − 1 2 D m f · g )] = 2 D x [( D t − 1 µ D z e − D m ) f · g ] · f g − 2 µ sinh( 1 2 D m ) n [( − µD z e 1 2 D m + λD z e − 1 2 D m − γ e 1 2 D m ) f · g )] · ( e − 1 2 D m f · g ) +[( λe − 1 2 D m + µe 1 2 D m ) f · g ] · ( D z e − 1 2 D m f · g ) o = 2 D x [( D t − 1 µ D z e − D m ) f · g ] · f g +2 sinh( 1 2 D m )[( D z e 1 2 D m f · g ) · ( e − 1 2 D m f · g ) − ( e 1 2 D m f · g ) · ( D z e − 1 2 D m f · g )] + 2 γ µ sinh( 1 2 D m )( e 1 2 D m f · g ) · ( e − 1 2 D m f · g ) = 2 D x [( D t − 1 µ D z e − D m ) f · g ] · f g +2 D z cosh( 1 2 D m )( e 1 2 D m f · g ) · ( e − 1 2 D m f · g ) + 2 γ µ sinh( 1 2 D m )( e 1 2 D m f · g ) · ( e − 1 2 D m f · g ) = 2 D x [( D t − 1 2 µ D z e − D m ) f · g ] · f g + γ µ 2 sinh( 1 2 D m )( D x e − 1 2 D m f · g ) · ( e − 1 2 D m f · g ) = 2 D x [( D t − 1 2 µ D z e − D m − γ 4 µ 2 e − D m ) f · g ] · f g = 0 Next, what we wan t to mention is that p ositive KP hiera rch y ma y b e derived fro m the sa me nonloc al symmetry (25 ) but with a different expansion. Actually we may consider the following situation: Case 3: λ = 0 , g = ∞ X i =0 g i µ − i , h = ∞ X i =0 h i µ − i . In this c ase, we hav e the following KP hiera rch y f t n − 1 = ( − 1) n 1 2 n 1 n ! f Z x  ∂ n ( gh ) ∂ µ n  | µ =0 f 2 dx ′ (60) ( D y + D 2 x + µD x ) f · g = 0 (61) ( D y + D 2 x + µD x ) h · f = 0 (62) 6 i.e. D x D t n − 1 f · f = ( − 1) n 1 2 n − 1 n X i =0 g i h n − i , (63) ( D y + D 2 x ) f · g i = − D x f · g i +1 , (64) ( D y + D 2 x ) h i · f = − D x h i +1 · f (65) with g 0 = h 0 = f . By direct calculations, we hav e, g 1 = 2 f x , g 2 = 2( − f y + f xx ) , g 3 = 8 3 f t 3 + 4 3 f xxx − 4 f xy , · · · and h 1 = − 2 f x , h 2 = 2( f y + f xx ) , h 3 = − 8 3 f t 3 − 4 3 f xxx − 4 f xy , · · · from which we ha ve D x D t 1 f · f = D 2 x f · f , D x D t 2 f · f = D x D y f · f which means we ma y choos e t 1 ≡ x and t 2 ≡ y . Remark In Sa to theor y , there is a famous genera ting f ormula for deriving all the bilinear equations o f the KP hierarchy [24]- [27] 3 Nonlo cal symmetries for the bilinear BKP equation and its ap- plication In this sectio n, we will cons ider nonlo c a l symmetry for bilinear BKP equation and its application to gener- ating nega tive and po sitive BKP hierar chies. The bilinear BK P reads [28] ( D 6 x − 5 D 3 x D y − 5 D 2 y + 9 D x D t ) f · f = 0 . (66) Its bilinea r BT is given as follows: ( D 3 x − D y ) f · g = 0 , (67) ( D 5 x + 5 D 2 x D y − 6 D t ) f · g = 0 , (68) or equiv a lently ( D 3 x − D y ) f · h = 0 , (69) ( D 5 x + 5 D 2 x D y − 6 D t ) f · h = 0 . (70) W e hav e the following result [10] Prop ositi on 3. σ given by σ = f Z x D x g · h f 2 dx ′ is a nonlo c al symmetry for the BKP e quation (66), i.e. σ satisfies symmetry e quation ( D 6 x − 5 D 3 x D y − 5 D 2 y + 9 D x D t ) σ · f = 0 . (71) wher e g and h satisfy (67)-(70). If g − → e λy g , h − → e − λy h , we hav e ( D 3 x − D y + λ ) f · g = 0 (72) ( D 5 x + 5 D 2 x D y − 6 D t − 5 λD 2 x ) f · g = 0 (73) ( D 3 x − D y − λ ) f · h = 0 (74) ( D 5 x + 5 D 2 x D y − 6 D t + 5 λD 2 x ) f · h = 0 (7 5) 7 and σ = f Z x D x g · h f 2 dx ′ is a nonlocal symmetry . In the following, w e would like to deriv e a hierarchy o f negativ e BKP equations. F or this purp o se, by considering g = ∞ X i =0 g i λ i , h = ∞ X i =0 h i λ i , we ma y write down the following negativ e BKP hiera r ch y f t − 3 n − 1 = 1 2 1 n ! f Z x  ∂ n ( D x g · h ) ∂ λ n  | λ =0 f 2 dx ′ (76) ( D 3 x − D y + λ ) f · g = 0 (77) ( D 3 x − D y + λ ) h · f = 0 (78) i.e. D x D t − 3 n − 1 f · f = n X i =0 D x g i · h n − i (79) ( D 3 x − D y ) f · g i + f g i − 1 = 0 ( 80) ( D 3 x − D y ) h i · f + h i − 1 f = 0 (81) with g − 1 = h − 1 = 0. Again, in consideration of the fact that there is only one τ function appe ared in the bilinear for m for p o sitive BKP hiera rch y , it is natural to inquire as to how to rewrite (79)-(81) int o a set of bilinear equations with o nly o ne τ function. In the follo wing, we will giv e some illustrative exa mples to show that b y in tr o ducing a sequence of a dditional v ariables m, z 1 , z 2 , · · · , (79)-(81) ma y b e transfor med into a set of bilinea r equatio ns with one dep endent v ariable f . Example 6: n=0. In this ca se, w e set f = f ( m ) , g 0 = f ( m − 1) , h 0 = f ( m + 1 ). Then t − 1 -flow of the NBKP hier a rch y (79)-(81) beco mes D x D t − 1 f · f = − D x e D m f · f (82) ( D 3 x − D y ) e 1 2 D m f · f = 0 . (83) Example 7: n=1. In this cas e , we set f = f ( m ) , g 0 = f ( m − 1) , h 0 = f ( m + 1) , g 1 = − f z 1 ( m − 1) , h 1 = f z 1 ( m + 1) . Then t − 2 -flow of the NBKP hierarchy (7 9)-(81) b ecomes D x D t − 4 f · f = − D x D z 1 e D m f · f (84) ( D 3 x − D y ) e 1 2 D m f · f = 0 , (85) D z 1 ( D 3 x − D y ) e 1 2 D m f · f + 2 e 1 2 D m f · f = 0 . (86) In general, along this line, we may rewr ite the NBKP hiera rch y (79)-(81) in terms of o ne τ function, step b y step. On the other hand, if g − → e kx + k 3 y + k 5 t g , h − → e − kx − k 3 y − k 5 t h , we hav e fro m (67)-(70) that ( D 3 x − D y − 3 k D 2 x + 3 k 2 D x ) f · g = 0 (87) ( D 5 x + 5 D 2 x D y − 6 D t − 5 k D 4 x + 5 k 2 D 3 x + 10 k 2 D y − 10 k D x D y ) f · g = 0 (88) ( D 3 x − D y − 3 k D 2 x + 3 k 2 D x ) h · f = 0 (89) ( D 5 x + 5 D 2 x D y − 6 D t − 5 k D 4 x + 5 k 2 D 3 x + 10 k 2 D y − 10 k D x D y ) h · f = 0 , (90) and σ given by σ = f Z x ( D x + 2 k ) g · h f 2 dx ′ (91) 8 is a nonlo cal symmetry . In this case, by expanding g and h a s follows g = ∞ X i =0 g i k i , h = ∞ X i =0 h i k i , we ma y write down another neg ative BKP hierarch y f t − n − 1 = 1 2 1 n ! f Z x  ∂ n (( D x +2 k ) g · h ) ∂ k n  | k =0 f 2 dx ′ (92) ( D 3 x − D y − 3 k D 2 x + 3 k 2 D x ) f · g = 0 (93) ( D 3 x − D y − 3 k D 2 x + 3 k 2 D x ) h · f = 0 (94) i.e. D x D t − n − 1 f · f = n X i =0 D x g i · h n − i + 2 n − 1 X i =0 g i h n − 1 − i (95) ( D 3 x − D y ) f · g i − 3 D 2 x f · g i − 1 + 3 D x f · g i − 2 = 0 (96) ( D 3 x − D y ) h i · f − 3 D 2 x h i − 1 · f + 3 D x h i − 2 · f = 0 (97) with g − 2 = g − 1 = 0 , h − 2 = h − 1 = 0. In the following, we wan t to show you how to rewr ite (95)-(97) into bilinea r equatio ns with only o ne τ function thro ug h some illustrative examples: Example 8: n=0. In this ca se, w e set f = f ( m ) , g 0 = f ( m − 1) , h 0 = f ( m + 1 ). Then t − 1 -flow of the NBKP hier a rch y (95)-(97) beco me D x D t − 1 f · f = − D x e D m f · f , (98 ) ( D 3 x − D y ) e 1 2 D m f · f = 0 (99) which coincides with (82) and (83). Concerning (98) and (99), we ha ve the following result: Prop ositi on 4. A B¨ acklund tr ansformation fo r (98) and (99) is ( D x e D m 2 − λD x e − D m 2 − µe D m 2 + λµe − D m 2 ) f · g = 0 , (100) ( D 3 x − D y − 3 µD 2 x + 3 µ 2 D x + γ ) f · g = 0 , (101) ( D t − 1 + 1 2 λ e D m − λ 2 e D m + ζ ) f · g = 0 (10 2) wher e λ, µ, γ and ζ ar e arbitr ary c onstants. Pr o of. Let f ( m ) b e a s o lution of Eqs. (98) and (99). If we can show that E qs. (1 00)-(102) guara ntee that the following tw o relations: P 1 ≡ ( D x D t − 1 + D x e D m ) g · g = 0 , (103) P 2 ≡ ( D 3 x − D y ) e 1 2 D m g · g = 0 , (10 4) hold, then Eqs. (1 0 0)-(102) form a B ¨ ack l und trans formation. In analogy with the proo f already g iven in [29], w e know that P 2 = 0 holds. Th us it suffices to show that P 1 = 0. In this regard, by using (A6)-(A7), we ha ve − P 1 f 2 = 2 D x ( D t − 1 f · g ) · f g + 2 sinh( 1 2 D m )[( D x e 1 2 D m f · g ) · ( e − 1 2 D m f · g ) − ( e 1 2 D m f · g ) · ( D x e − 1 2 D m f · g )] = 2 D x ( D t − 1 f · g ) · f g + 2 sinh( 1 2 D m )[( D x e 1 2 D m − λD x e − 1 2 D m ) f · g ] · [( − 1 λ e 1 2 D m + e − 1 2 D m ) f · g ] +2 sinh( 1 2 D m )[ 1 λ ( D x e 1 2 D m f · g ) · ( e 1 2 D m f · g ) + λ ( D x e − 1 2 D m f · g ) · ( e − 1 2 D m f · g )] = 2 D x ( D t − 1 f · g ) · f g + 1 λ D x ( e D m f · g ) · f g − λD x ( e − D m f · g ) · f g = 0 . 9 Example 9: n=1. In this cas e , we set f = f ( m ) , g 0 = f ( m − 1) , h 0 = f ( m + 1) , g 1 = − f z 1 ( m − 1) , h 1 = f z 1 ( m + 1) . Then t − 2 -flow of the NBKP hierarchy (3 0)-(32) b ecomes D x D t − 2 f · f = ( − D x D z 1 e D m + 2 e D m ) f · f (105) ( D 3 x − D y ) e 1 2 D m f · f = 0 , (106) D z 1 ( D 3 x − D y ) e 1 2 D m f · f − 6 D 2 x e 1 2 D m f · f = 0 . (107) Example 10: n=2. In this ca s e, we set f = f ( m ) , g 0 = f ( m − 1) , h 0 = f ( m + 1) , g 1 = − f z 1 ( m − 1) , h 1 = f z 1 ( m + 1) , g 2 = − f z 2 ( m − 1) + 1 2 f z 1 z 1 ( m − 1) , h 2 = f z 2 ( m + 1) + 1 2 f z 1 z 1 ( m + 1) Then t − 3 -flow of the NBKP hierarchy (3 0)-(32) b ecomes D x D t − 3 f · f = ( − D x D z 2 e D m − 1 2 D x D 2 z 1 e D m + 2 D z 1 e D m ) f · f (108) ( D 3 x − D y ) e 1 2 D m f · f = 0 , (109) D z 1 ( D 3 x − D y ) e 1 2 D m f · f − 6 D 2 x e 1 2 D m f · f = 0 , (110) [ D z 2 ( D 3 x − D y ) e 1 2 D m + 1 2 D 2 z 1 ( D 3 x − D y ) e 1 2 D m − 3 D 2 x D z 1 e 1 2 D m + 6 D x e 1 2 D m ] f · f = 0 . (111) In ge ne r al, along this line, we may r ewrite the NBKP hierarch y (79)-(81) in ter ms o f o ne τ function, step by step. Finally , similarly as in section 2 , we want to mention that po sitive BKP hierar chy may b e derived fro m the same nonlocal symmetry (91) but with a differen t expansion. Actually w e may consider the following situation: Case 3: g = ∞ X i =0 g i k − i , h = ∞ X i =0 h i k − i . In this c ase, we hav e the following BKP hier a rch y f t n − 1 = 1 4 ( − 1) n 1 n ! f Z x  ∂ n (( D x +2 k ) g · h ) ∂ k n  | k =0 f 2 dx ′ (112) ( D 3 x − D y − 3 k D 2 x + 3 k 2 D x ) f · g = 0 (113) ( D 3 x − D y − 3 k D 2 x + 3 k 2 D x ) h · f = 0 (114) i.e. D x D t n − 1 f · f = 1 2 n X i =0 D x g i · h n − i + n +1 X i =0 g i h n +1 − i (115) ( D 3 x − D y ) f · g i − 3 D 2 x f · g i +1 + 3 D x f · g i +2 = 0 (116) ( D 3 x − D y ) h i · f − 3 D 2 x h i +1 · f + 3 D x h i +2 · f = 0 (117) with g 0 = h 0 = f . By direct calculations, we hav e, g 1 = − 2 f x , h 1 = 2 f x , g 2 = h 2 = 2 f xx , g 3 = − 2 3 f y − 4 3 f xxx , h 3 = 2 3 f y + 4 3 f xxx , g 4 = h 4 = 4 3 f xy + 2 3 f xxxx , · · · from which we ha ve D x D t 1 f · f = D 2 x f · f D x D t 2 f · f = 0 , D x D t 3 f · f = D x D y f · f which means we ma y choos e t 1 ≡ x and t 3 ≡ y , and f t 2 = 0, i.e. f is t 2 -indep endent. 10 4 Conclusion and discussions In this pap er, w e hav e in vestigated nonlo cal symmetries for the bilinea r KP and bilinear BKP equations. By expanding these nonlo cal symmetries, we ha ve deriv ed tw o types of bilinear NKP hierarchies and tw o t ype s of bilinear NBKP hierarchies. Interesting thing is that bilinear positive and negative KP and BKP hierarchies may b e derived from the same nonlo cal symmetrie s for the KP and BKP equatio ns. It s till remain unclea r what k ind o f explicit relations will ex is t b etw een the obtained tw o NKP hie r archies or tw o NBKP hierarchies. Our study strong ly sugg ests that these obtained NKP hierarchies and NBK P hierarchies should have many nice integrable prop erties. F or example, we hav e given a bilinear BT for t − 2 -flow of the NKP hierar ch y (45)-(47) and a bilinear B T for t − 1 -flow o f the NBKP hierarch y (98) a nd (99). W e can also co nsider bilinear BTs for o ther members of these NKP and NBKP hierarchies. F urthermore, using BT (53)-(55) and BT (100)-(102), w e can obtain soliton solutions for t − 2 -flow of the NKP hierarch y (4 5 )-(47) and t − 1 -flow (98)-(99) of the NBKP hierarch y . As for structures of τ functions for these NKP and NBKP hierarchies, further w or k needs to be done. Besides, from Prop o sition 3, w e know that in paticular , if we choose h = f , w e ha ve σ = g is a nonlo cal symmetry for the BKP equation. Then Σ = g + C xf is also a symmetry for the BKP equation (66). In this case, w e ma y derive the follo wing negativ e BKP equa tion fr om the symmetr y Σ: f t − 1 = g + C xf , (118) ( D 3 x − D y ) f · g = 0 (119) from which we ha ve [ D t − 1 ( D y − D 3 x ) + 6 C D 2 x ] f · f = 0 . If C = 1 2 , then [ D t − 1 ( D y − D 3 x ) + 3 D 2 x ] f · f = 0 which coincides with B K P − 1 given in Hirota’s b o o k [21]. Ac kno wledgemen ts This work was supp orted by the Nationa l Natural Science F oundation of China (grant no s . 10771 207, 10735 030 and 9050 3006). App endix A. Hirota bilinear op erator ident ities. The following bilinear o p erator identities hold for a rbitrary functions a , b , c , and d . ( D x D t a · a ) b 2 − a 2 D x D t b · b = 2 D x ( D t a · b ) · ab. (A1) ( D z e D m a · a ) b 2 − a 2 D z e D m b · b = 2 D z cosh( 1 2 D m )( e 1 2 D m a · b ) · ( e − 1 2 D m a · b ) . 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